Abstract fractional linear transformations

TL;DR

This paper explores fractional linear transformations on Banach algebras, leading to new insights into noncommutative polynomials, ring structures, and properties of projective elementary groups, with implications for algebraic stability and simplicity.

Contribution

It introduces a novel framework for FLT on rings, defines a length function on PE(2,R), and proves new results on the structure and simplicity of these groups.

Findings

Established invertibility properties of noncommutative polynomials.

Defined a length function on PE(2,R) and related it to stable range conditions.

Proved that under certain conditions, the commutator subgroup of PE(2,R) is perfect and often simple.

Abstract

We begin with (densely-defined) fractional linear transformations (FLT) on (some) Banach algebras and their relatives. This leads to Wedderburn's continued fractions (recursively-defined noncommutative polynomials) for any ring. Along the way, we discover a one-parameter family of (noncommutative) polynomials \st if one of them is invertible, then read in the opposite order, the corresponding polynomial is also invertible (extending the well known $1+ab$ is invertible if $1+ ba$ is, and the not-so-well-known, $a + abc + c$ and $a + cba + c$). This in turn leads to a definition of FLT for general rings $R$, which turns out to be PE$(2,R)$ (the projective elementary group). Using Wedderburn's polynomials, this permits us to define a length function on PE$(2,R)$, which suggests a stable range type condition (for $n =1$, it {\it is\/} stable range one, but higher values do not correspond. Again using the length results, we prove the expected results for PE$(2,R)$: under very modest conditions on $R$, the commutator subgroup of PE$(2,R)$ is perfect and of index one or two. Along the same lines, we also prove results on simplicity of the commutator subgroup: we require the usual generative properties on the simple ring $R$, as well either the very restrictive $1$ in the range, or a mild condition about invertibles, involving intersections of three translates of GL$(1,R)$. This last property is explored in the appendices, which give examples (and non-examples). Numerous questions suggest themselves throughout.